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Mirrors > Home > MPE Home > Th. List > ndmovg | Structured version Visualization version GIF version |
Description: The value of an operation outside its domain. (Contributed by NM, 28-Mar-2008.) |
Ref | Expression |
---|---|
ndmovg | ⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 6977 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | eleq2 2847 | . . . . . 6 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (〈𝐴, 𝐵〉 ∈ dom 𝐹 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆))) | |
3 | opelxp 5439 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) | |
4 | 2, 3 | syl6bb 279 | . . . . 5 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (〈𝐴, 𝐵〉 ∈ dom 𝐹 ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆))) |
5 | 4 | notbid 310 | . . . 4 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 ↔ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆))) |
6 | ndmfv 6526 | . . . 4 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → (𝐹‘〈𝐴, 𝐵〉) = ∅) | |
7 | 5, 6 | syl6bir 246 | . . 3 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐹‘〈𝐴, 𝐵〉) = ∅)) |
8 | 7 | imp 398 | . 2 ⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐹‘〈𝐴, 𝐵〉) = ∅) |
9 | 1, 8 | syl5eq 2819 | 1 ⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∅c0 4172 〈cop 4441 × cxp 5401 dom cdm 5403 ‘cfv 6185 (class class class)co 6974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-xp 5409 df-dm 5413 df-iota 6149 df-fv 6193 df-ov 6977 |
This theorem is referenced by: ndmov 7146 curry1val 7606 curry2val 7610 1div0 11098 repsundef 13988 cshnz 14012 cshnzOLD 14013 mamufacex 20717 mavmulsolcl 20879 mavmul0g 20881 iscau2 23598 1div0apr 28039 rrxsphere 44137 |
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